Csci 6212 design and analysis of algorithms dynamic programming
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CSCI 6212 Design and Analysis of Algorithms Dynamic Programming. Dr. Juman Byun The George Washington University. Please drop this course if you have not taken the following prerequisite. Sometimes enthusiasm alone is not enough. CSci 1311: Discrete Structures I (3)

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CSCI 6212 Design and Analysis of Algorithms Dynamic Programming

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CSCI 6212 Design and Analysis of AlgorithmsDynamic Programming

  • Dr. Juman Byun

  • The George Washington University

  • Please drop this course if you have not taken the following prerequisite. Sometimes enthusiasm alone is not enough.

  • CSci 1311: Discrete Structures I (3)

  • CSci 1112: Algorithms and Data Structures (3)


n=4

Example: Rod Cutting


n=4

Example: Rod Cutting

Maximum Revenue, r4 ?


rn when n=4 ?

$10

$9

$1

$8

$5

$5

$8

$1

$1

$1

$5

$1

$5

$1

$5

$1

$1

$1

$1

$1

$1


Notation

$10

$5

$5

4-inch

rod into 2 pieces

Decomposition:

4 = 2 + 2

Maximum Revenue:

r4 = $5 + $5


Notation

rn

n-inch

rod into k pieces

Decomposition:

n = i1 + i2 + … + ik

Maximum Revenue:


r1 + rn-1

Uncut Rod of length n

pn

General Procedure to Find Optimal Rod Cutting

Cut

Revenue

Pick the largest

r2 + rn-2

rn-2 + r2

rn-1 + r1


General Procedure to Find Optimal Rod Cutting


General Procedure to Find Optimal Rod Cutting


Recursive Top-Down

  • Cut-Rod(p,n)

  • if n == 0

  • return 0

  • q = ∞

  • for i = 1 to n

  • q = max(q,p[i] + Cut-Rod(p, n - i ) )

  • return q


vs Divide-and-conquer

  • Similarity

  • to divides problems into subproblems

  • Difference

  • subproblems overlap


Can we do better ?


Momoized-Cut-Rod

  • Memoized-Cut-Rod(p,n)

  • let r[0..n] be a new array

  • for i = 0 to n

  • r[i] = -∞

  • return Memoized-Cut-Rod-Aux(p,n,r)


Momoized-Cut-Rod-Aux

  • Momoized-Cut-Rod-Aux(p,n)

  • if r[n] >= 0

  • return r[n]

  • if n == 0

  • q = 1

  • else q = -∞

  • for i = 1 to n

  • q = max(q,p[i]+Memoized-Cut-Rod-Aux(pn,n-i,r))

  • r[n] = q

  • return q


Bottom-Cut-Rod

  • Bottom-Up-Cut-Rod(p,n)

  • let r[0..n] be a new array

  • r[0] = 0

  • for j = 1 to n

  • q = -∞

  • for i = 1 to j

  • q = max(q, p[i] + r[j-i])

  • r[j] = q

  • return r[n]


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